optimal control problem for the nonlinear hyperbolic systems 1 ...

数理解析研究所講究録 1187 巻 2001 年 27-36

27

OPTIMAL CONTROL PROBLEM FOR THE NONLINEAR HYPERBOLIC SYSTEMS JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE

ABSTRACT. In this paper we study parameter optimal control monitored by nonlinear hyperbolic systems. We show that for every value of the parameter, the optimal control problem has a solution. Moreover we obtain the necessary optimality condition on the control system.

1.

INTRODUCTION

The optimal control problems have been extensively studied by many authors [1.3,5,7.10,13 and reference there in] and also identification problem for damping parameters in the second order hyperbolic systems have been dealt with by many authors [6,8,12 and there reference in]. In this paper, we consider the following control systems $y”+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=Bu+f\cdot(t, q)$

(1.1)

$\{$

$y(q, u)(0)=y_{\mathrm{U}}\in V,$

$y’(q, u)(0)=y_{1}\in H$

and the cost functional given by the quadratic form (1.2)

Here

$J(q, u)= \frac{1}{2}||Cy(q, u)-z_{d}||_{M}^{2}$

$A_{1}(t, q)$

, and

$A_{2}(t, q)$

.

are differential operators containing unknown parameter

and there are given by some bilinear forms on Hilbert spaces. term.

$B$

is a controller.

$u\in U$

is a control,

operator defined on an observation space

$f$

$M,$

is a forcing term and $z_{d}$

inf

$J(q, u)=J(\overline{q},\overline{u})$

. In this paper

$C$

is a nonlinear

is an observation

is a desired value. The optimal con-

trol problem subject to (1.1) nith (1.2) is to find an element $(q.u)\in Q\cross lI$

$N^{*}g(Ny)$

$q\in Q$

$(\overline{q},\overline{u})\in Q\cross U$

such that

we will study the optimal control to the system

(1.1) with (1.2) and the existence of weak solution for (1.1). It is not easy to find the optimal control pairs

$(\overline{q},\overline{\prime(x})$

belonging to a general admissible set

$Q\cross U$

of parameters and

controls subject to (1.1) with (1.2). Hence we will show the existence of such

$(\overline{q},\overline{u})$

when

1991 Mathematics Subject Classification. . Key words and phrases. Optimal control, nonlinear hyperbolic systems, quadratic cost function. This work was supported by Brain Korea 21. 1999. $93\mathrm{C}20,49\mathrm{J}20,49\mathrm{K}‘ \mathit{2}0$

28 JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE $Q\cross U$

is a compact subset of a topological space. Recently, inspired by the optimal con-

trol theoretical studies of Euler-Bernoulli Beam Equations with Kelvin-Voigt Damping, and Love-Kirchoff Plate Equations with various damping terms, the appeared numerous paper studying optimal control theory and identification problems. In Banks et al. [4],

Banks and Kunisch [5], they treated the existence of the optimal control (or minimizing parameters) by using the methods of approximations, but they didn’t deal with the nec-

essary conditions (or characterizations) on them. When $N^{*}g(Ny)=0$

$A_{1}(t, q)\equiv\gamma A_{2}(t, q),$ $\gamma>0$

in (1.1), the identification problem estimating

identification problem is studied by Ahmed

$[1,2]$

and

via output least-square

$q$

based on the transposition methods. In

the nonlinear parabolic type case, Papageorgiou [

$11_{\rfloor}^{-}$

treated with the optimal control

problems contained parameter and control. But we deal with the second order nonlinear hyperbolic systems.

In specially. in this paper we study the optimal control (or minimizing parameters) problems to (1.1) with (1.2) on the Gelfand five fold and the necessary conditions.

2. Let

$X$

be a real Hilbert spaces.

induced norm on X. between

$X^{*}$

and

$X$

$|\cdot|_{H}$

spaces

and

$||\cdot||_{X}$

$X$

and

denote the inner product and the denotes the dual pairing

,

$\langle\cdot, \cdot\rangle_{XX}.$

. Let us introduce underlying Hilbert spaces to describe the nonlinear $H$

be a real pivot Hilbert space, its norm

$||\cdot||_{H}$

is denoted simply

. Throughout this paper we assume there is a sequence of real separable Hilbert

$V_{1},$

$V_{2}^{*}arrow fV_{1}^{*}$

with

$(\cdot, \cdot)_{X}$

the dual space of

$X^{*}$

hyperbolic systems. Let by

PRELIMINARIES

$V_{2},$

$V_{1}^{*},$

$V_{2}^{*}$

forming a Gelfand quintuple satisfying

. And also we assume that the embedding

$||\phi||_{V_{2}}\leq c||\phi||_{V_{1}}$

now on, we write

for

$\phi\in V_{1}$

$V_{1}=V$

$\langle\phi, \varphi\rangle_{V^{*},V}=\langle\phi, \varphi\rangle_{V_{2}^{*},V_{2}}$

for

and

$V_{2}\mathrm{c}_{arrow}H$

$V_{1}arrow\rangle$

$V_{1}arrow\rangle$

$V_{2}$

$V_{2}arrow\rangle$

$H\equiv H^{*}arrow\rangle$

is dense and continuous

is a densely compact embedding. From

for convenient of notation. We assume that the equalities $\phi\in V_{2}^{*},$ $\varphi\in V$

and

$\langle\phi, \varphi\rangle_{\mathcal{V}^{*},V}=(\phi, \varphi)_{H}$

for

$\phi\in H,$

$\varphi\in V$

.

We shall give an exact description of the nonlinear hyperbolic systems. We suppose that $Q$

is algebraically contained in a linear topological vector space with topology

$Q_{\Gamma}.=(Q, \tau)$

space

$Y$

is compact. And also we suppose that

. Let $I=[0, T],$ $T\geq 0$ be fixed and

$t\in[0, T]$

$U$

and

is compact subspace of Hilbert

. Let

We will need the following hypotheses on the data.

$\tau$

$q\in Q_{\tau}$

.

29 OPTIMAL CONTROL PROBLEM FOR THE HYPEMBOLIC SYSTEMS. $\mathrm{H}(\mathrm{A})$

(1)

:

:

$A_{i}$

$I\cross Qarrow \mathcal{L}(V_{i}, V_{i})$

is an operator $(i=1,2)$ .

$a_{i}(t, q;\phi, \varphi)=a_{i}(t, q;\varphi, \phi)$

, where

(2) There exists

$c_{\vee i1}>0$

such that

(3) There exists

$\alpha_{i}>0$

and

(4) The function (5) There exists

and $\mathrm{H}(\mathrm{f})$

$\mathrm{H}(\mathrm{B})$

$\mathrm{H}(\mathrm{N})$

$f$

:

$B:Yarrow V_{2}^{*}$

:

$N$

:

such that

:

$I\cross Qarrow V_{2}^{*}$

:

$V_{i}arrow H$

:

$Harrow H$

$\varphi\in V_{i}$

$|a_{i}’(t, q;\phi, \varphi)|\leq c_{i2}||\phi||_{V_{i}}||\varphi||v_{i},$

$\forall\phi,$

$\forall\phi\in V_{i}$

.

.

$\varphi\in V_{i},$

where

$’= \frac{d}{d\mathrm{f}}$

.

is the forcing term such that

$f(t, q)\in L^{2}(0, T;V_{2}^{*})$

is a bounded linear operator such that is a linear operator such that $N$

$[0, T]$

.

.

$a_{i}(t, q;\phi, \varphi)+\lambda_{i}|\phi|_{H}^{2}\geq\alpha_{i}||\phi||_{V_{i}}^{2},$

is continuously differentiable in

is constant and the range of $g$

such that

$a_{i}’(t, q:\phi, \varphi)=\langle A_{i}’(t, q)\phi, \varphi\rangle_{V_{i}}*,v_{i}$

:

$k_{i}$

$\mathrm{H}(\mathrm{g})$

.

$c_{i2}>0$

$\forall\phi,$

$|a_{i}(t, q;\phi, \varphi)|\leq c_{i1}||\phi||v_{i}||\varphi||v_{i},\forall\phi,$ $\varphi\in V_{i}$

$\lambda_{i}\in R$

$t\mapsto a_{i}(t, q;\phi, \varphi)$

$a_{i}(t, q;\phi, \varphi)=\langle A_{i}(t, q)\phi, \varphi\rangle v_{i}*,v_{i},$

on

$V_{i}$

$B\in L^{\infty}(\mathrm{O}, T;\mathcal{L}(Y, V_{2}^{*}))$

$N\in \mathcal{L}(V_{\mathrm{i}}, H)$

is dense in

$H$

.

with

.

$||N\varphi||\leq\sqrt{k_{i}}||\varphi||_{V_{2}}$

,

.

is a continuous nonlinear mapping of real gradient

$(\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l})\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}$

such that (1)

$||g(\varphi)||\leq c_{1}||\varphi||+(- 2,$

(2)

$||g(\varphi)-g(\phi)||\leq C_{\backslash }‘’||\varphi-\phi||,$

$\varphi\in H$

and for some constant $\varphi,$

$\phi\in H$

$c_{1},$

$\mathrm{c}_{-2}$

,

and for some constant

$C’.;$

.

We consider the following problem for the nonlinear second order evolution equations of the form : (2.1)

$y”+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=f(t, q)$

(2.2)

where

$y(q)(0)=.\tau/\{)\in V,$

$y’= \frac{d?/}{dt},$ $y”= \frac{d^{2}y}{dt^{2}}$

$y’(q)(0)=y_{1}\in H$ ,

. We define a Hilbert space. which will be a space of solutions.

as $W(0, T)=\{y|y\in L^{2}(0, T;V), y’\in L^{2}(0, T:V_{2}), y’’\in L^{2}(0, T;V^{*})\}$

with an inner product $(y_{1},.y_{2})_{W((),T)}= \int_{0}^{T}\{(.\tau/\iota(t), y_{2}(t))_{\mathcal{V}}+(y_{1}’(t,),y_{\mathit{2}}’‘(t))_{V_{2}}+(y_{1}’’(t), y_{2}’’(t))_{V^{*}}\}dt$

and the induced norm $||y||_{W((),T)}=(||_{lj}’.||^{\frac{")}{L}}9((),\tau);V)+||_{l/’}’.||_{L^{9}((),T;1^{r_{9}})}^{2}.+||y^{\prime/}||_{L^{2}((\mathrm{I},T;V^{*})}^{2})^{\frac{1}{2}}$

We denote by

$D(\mathrm{O}, T)$

the space of distributions on

$(0, T)$ .

.

30 JONG YEOUL PARK, YONG HAN KANG AND MI JIN LEE

Definition 2.1. A function and

$y$

(2.3)

is said to be a weak solution of

$(2.1)-(2.2)$

if $y\in

W(0, T)$

satisfies $\langle y^{\prime/}(\cdot), \phi\rangle_{V^{*},V}+a_{2}(\cdot)q;y’(\cdot),$

for all (2.4)

$y$

$y(q)(0)=y_{()}\in V,$

$\phi)+a_{1}(\cdot, q;y(\cdot),$ $\phi)+\langle g(N.y(\cdot)), N\phi\rangle_{H}=\langle].(\cdot, q), \phi\rangle_{V_{2}^{*},V_{2}}$

$\phi\in V$

in the sense of

$\frac{dy}{db}(q)(0)=y_{1}\in H$

$D(\mathrm{O}, T)$

.

By Definition2.1 it is verified that a weak solution (2.5)

,

of (2.1) satisfies

$y$

$\int_{()}^{T}\langle y’’(t)+A_{2}(t, q)y’(t)+A_{1}(t, q)y(t)+N^{\lambda}g(Ny(t,)), \phi(t)\rangle_{\mathrm{I}/V_{2}}2^{\cdot}’ dt$

$= \int_{()}^{T}\langle f(t, q), \phi(t,)\rangle_{V_{2}^{*},V_{2}}dt,$

$\forall\phi)\in L^{2}(0, T;V_{2})$

We state the existence and uniqueness results of a weak solution of Theorem 2.1.

If $H(A),H(f),H(B),H(N)$ and $H(g)$ hold and

.

$(2.1)-(2.2)$ .

$L(t)sat,isfyL(\cdot)\in L^{\infty}(\mathrm{O}, T;\mathcal{L}(V_{2}, V_{2}^{*}))$

Then the equat,ion (2.6)

$\{$

$y”+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=L(t)y+f(t, q)$ $y(q)(0)=?/0\in V,$ $y’(q)(0)=y_{1}\in H$ ,

has a unique we $aksolut,iony\in weak solution for (2.6)

$\dot{w}$

W(0, T)\cap C(\mathrm{O}, T;V)\cap C^{1}(0, T;H)$

defined

init, $ial$

conditions

$Her\cdot et,ht_{J}^{I}$

,

concep , $t$

of a

$\phi)+a_{1}(\cdot, q;y(\cdot),$ $\phi)+\langle g(Ny(\cdot)), N\phi\rangle_{H}$

$=\langle L(\cdot)y(\cdot)+f(\cdot, q), \phi\rangle_{V_{2}V_{\underline{9}}}.,,$ $\forall\phi\in V$

the

$(0, T)$

as

$\langle y’’(\cdot), \phi\rangle_{V^{*},V}+a_{2}(\cdot, q;y’(\cdot),$

$wit,h$

.

in

in

$t,h(^{\lrcorner}$

,

sens

$\rho_{J}$

of $D’(0, T)$

$y(q)(0)=y_{0}\in V,$ $y’(q)(0)=.l/1\in H$ .

PROOF. We can prove by using the method Lions [9] and Ha [8]. 3. EXISTENCE OF

BOTH PARAMETERS AND CONTROLS FOR OPTIMALITY

In this section we consider the optimal control problem for the following system: (3.1)

$y^{\prime/}+A_{2}(t, q)y’+A_{1}(t, q)y+N^{*}g(Ny)=Bu+f(t, q)$ $\{$

$y(q, u)(0)=y_{()}\in V,$ $y’(q, u)(0)=y_{1}\in H,$

Note that since there is a unique solution well-defined mapping $y=y(q, u)$ of

$y$

$Q_{\tau}\cross[I$

$q\in Q_{\tau},$

to (3.1) for given

into

$W(0, T)$ .

in .

$(0, T)$

$u\in[I$

$(q, u)\in Q_{\tau}\cross[’$

, we have a

31 OPTIMAL CONTROL PROBLEM FOR THE HYPERBOLIC SYSTEMS.

We often call (3.1) the state equation and

$y(q, u)$

the state with respect to (3.1). Let us

consider a quadratic cost functional attached to (2.6) as (3.2)

$J(q, u)= \frac{1}{2}||Cy(q, u)-z_{d}||_{M}^{2},$ $(q, u)\in Q_{\tau}\cross U$

where

is a Hilbert space of observations,

$M$

a desired value belonging to (3.3)

$M$

.

is an observer and

$C\in \mathcal{L}(W(0, T),$ $M)$

Our main aim is to find

satisfying

$(\overline{q},\overline{u})\in Q_{\tau}\cross[\Gamma$


. We call


the optimal control to the

system (3.1) and (3.2). Furthermore. we will give an assumption to

$H(A)_{1}$

is

$J( \overline{q},\overline{u})=\min_{(q,u)\in Q_{\tau}\cross U}J(q, u)$

and to give a characterization of such

and

$z_{d}$

$f\cdot$

:

$a_{i}(t, q:\phi, \varphi),$

$i=1,2$

: $qarrow a_{i}(t, q;\phi, \varphi)$

Note that for each

:

$Q_{\tau}arrow R$

$q\in Q_{\tau},$

$\phi,$

is continuous for all

$\varphi\in V_{i}$

$f,$

$\in[0, T],$

$\phi,$

$\varphi\in V_{i}$

.

the following equalities hold :

$|| \sup_{\varphi||_{V_{i}}=1}|a_{i}(t, q;\phi, \varphi)|=\sup_{||\varphi||_{V_{i}}=1}|\langle A_{i}(t, q)\phi, , \varphi\rangle_{\iota/V_{i}}*.|=||A_{i}(t, q)\phi||_{V_{i}^{*}}$

whence the assumption

$H(A)_{1}$

and the above equality imply that

,

$||A_{\mathrm{i}}(t, q)\phi||_{V^{*}}$

,

is contin-

uous on . $q$

$H(f)_{1}$

:

$qarrow f\cdot(\cdot, q)$

:

$Q_{\tau}arrow V_{2}^{*}$

is continuous.

Lemma 3.1. If $H(A),H(f),H(B),H(N),H(A)_{1}$ and $L^{\infty}(0, T;\mathcal{L}(V_{2}, V_{2}^{*}))$

$U,$

$W(0, T))$

. Then

$y(q, u)$

is

$H(f\cdot)_{1}$

$str\cdot onglycont,inuo\uparrow\iota s$

hold and also $L(t)$ satisfy on

$L(\cdot)\in$

$(q, u),$ $i.e.,$ $y(q, u)\in C(Q_{\tau}\cross$

.

PROOF. It can be proved by using the method of

Theorem 3.1.

If

$H(A),H(f),H(B)_{\rangle}H(N),H(A)_{1}$

$L(\cdot)\in L^{\infty}(\mathrm{O}, T;\mathcal{L}(V_{2}, V_{2}^{*}))$

. Then there

$?S$

$\mathrm{A}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{d}[2]$

and

$H(f)_{1}$

and

$\mathrm{H}\mathrm{a}[8]$

.

hold and also $L(t)sat,isfy$

at least one optimal control


if

$Q_{\tau}\cross$

[’

$is$

compact.

PROOF. It is clear from Lemma 3.1 and continuity of norm.

4. NECESSARY CONDITION

$\square$

OF OPTIMALITY FOR BOTH PARAMETERS AND CONTROLS

Here we present the necessary condition (the minimizing condition) for the optimal controls

$(\overline{q},\overline{u})\in Q_{\tau}\cross U$

to the system (3.1) with the cost functional

$J(p, u)$

given by


(3.2). If

$J(p,u)$

is G\âteaux differentiable at

necessary condition on (4.1)

,

in the direction


.

Note that since

$J(q, u)$

composed of the term

follows ffom that of

$J(q, u)$

the

,

$\forall(q, u)\in Q_{\tau}\cross U$

denotes the G\âteaux derivative at

$DJ(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})$

$(q-\overline{q}, u-\overline{u})$


is characterized by the following inequality


$DJ(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})\geq 0,$

where

in the direction


$y(q, u)$ .

$y(q, \prime\prime x)$

, the G\âteaux differentiability of

Hence to obtain that of

$y(q, u)$

we will need the

following condition: $H(A)_{2}$

:

$qarrow A_{i}(\cdot, q)$

is G\âteaux differentiable for all

$L^{2}(0, T;\mathcal{L}(V_{i}, V_{i}^{*}))$

at $H(g)_{1}$

for all

$q\in Q_{\tau}$

, where

$f_{J}$

and

$DA_{i}(t, q:p)$

$DA_{i}(t, q)(p)\equiv DA_{i}(t, q;p)\in$

denotes the G\âteaux derivative

in the direction of . $p$

$q$

: For any with

$\varphi\in H$

the R\’echet derivative of

$||g_{\varphi}(\varphi)||_{\mathcal{L}(H,H)}\leq c_{4}$

, where

$g_{\varphi}(\varphi)$

$g$

exists and satisfies

$g_{\varphi}(\varphi)\in \mathcal{L}(H, H)$

is the R\’echet derivative of

at

$g$

$\varphi$

and

(

$i_{4}$

is

constant. $H(f)_{2}$

:

$qarrow f(t, q)$

where

is G\âteaux differentiable for all and $f_{q}(t, q)p\equiv f_{q}(t, q;p)\in L^{2}(0, T, V_{2}^{*})$ , $t$

$f_{q}(t, q;p)$

is G\âteaux derivative at

$q$

in the direction of . $p$

Lemma 4.1. Assume that the conditions in Theorem 2.1, and

$H(g)_{1}$

direction which

satisfied.

aoe

$(q-\overline{q}, u-\overline{u}),$

satisfies

Then

$y(q, u)$

is weakly G\ât,eaux

$der\iota ot,et,heG\hat{a}tea^{l}\llcorner\iota x$

derivative

$H(A)_{1},$ $H(A)_{2},$

differentiable

of $y(q, u)$ by

$H(f\cdot)_{1},$

$af,$

$(q, u)$

$H(f\cdot)_{2}$

in the

$z=Dy(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})$

,

the following Cauchy problem: $z^{\prime J}+A_{2}(t,\overline{q})z’+A_{1}(t,\overline{q})z+N^{*}g_{y}(Ny(\overline{q},\overline{u}))Nz$ $=-DA_{2}(t,\overline{q};q-\overline{q})y’(\overline{q},\overline{u})-DA_{1}(t,,\overline{q};q-\overline{q})y(\overline{q},\overline{u})$

(4.2)

$\{$

$+B(u-\overline{u})+f_{q}(t,\overline{q};q-\overline{q})$

in

$(0, T)$

$z(0)=z’(0)=0$ .

PROOF. We can prove by using the method of Ahemd [2] and Park et al. [12]. By Lemma 4.1, the cost functional

direction (4.3)


$J(q, u)$

is G\âteaux differentiable at


$\square$

in the

, and so, the condition (4.1) is rewritten by

$DJ(\overline{q},\overline{u};q-\overline{q}, u-\overline{u})=\langle C^{*}\Lambda_{M}(C.y(\overline{q},\overline{u})-z_{d}).z\rangle_{W^{*}((),T),W((),T)}$

$+\langle C^{*}\Lambda_{M}(Cy(\overline{q},\overline{?x})-z_{d}), y_{u}(\overline{q},\overline{u};u-\overline{u})\rangle_{W^{*}(\mathrm{U}’I’),W(\mathrm{U},T)},\geq 0,$

$\forall(q)u)\in Q_{\tau}\cross[’$

,


where of

$C$

(i)

$z$

is a unique weak solution to (4.2),

and

$\Lambda_{M}$

is the canonical isomorphism of

$\langle\Lambda_{M}\phi, \phi\rangle_{M^{\kappa},M}=||\phi||_{M}^{2}$

(ii)

$C^{*}\in \mathcal{L}(M^{\lambda}, W^{*}(\mathrm{O}, T))$

$||\Lambda_{M}\phi||_{M}\cdot=||\phi||_{M}$

$M$

onto

$M^{*}$

is the adjoint operator

in the sense that

,

for all

$\phi\in M$

. In order to avoid the complexity of setting up

observation spaces, we consider the following two types of distributive and terminal value observations in time sense. that is, the following cases : (i) we take (ii) we take

$C_{1}\in \mathcal{L}(L^{2}(0, T;V_{2}),$

$C_{2}^{\mathrm{v}}\in \mathcal{L}(H, M)$

4.1. The case where

$M)$

and observer

and observer

$z(q, \prime n)=C_{1}y(q, u)$

$z(q, u)=C_{2}y(q, u)(T)$

$C_{1}\in \mathcal{L}(L^{\underline{y}}‘(0, T;V_{2}),$

;

.

$M)$

In this case the cost functional is given by $J(q, u)= \frac{1}{2}||C_{1}y(q, u)-z_{d}||_{M}^{2},$ $\forall

q\in Q_{\tau}\mathrm{x}U$

,

and then the necessary condition (4.3) is equivalent to (4.4)

$\int_{\{)}^{T}\langle C_{1}^{*}\Lambda_{M}(C_{1}y(\overline{q}, ’\overline{\iota x})(t)-z_{d}), z(t)\rangle_{V_{2}^{*},V_{2}}dt$

$+ \int_{\{\}}^{T}\langle C_{1}^{*}\Lambda_{M}(C\mathrm{i}/(\overline{q},\overline{u})(t)-z_{d}), .y|J(\overline{q},\overline{u};u-^{r}\overline{(x})\rangle_{1^{\gamma*}V_{2}},\rangle’ dt\geq 0,$

Let us introduce an adjoint state (4.5)

$\eta(\overline{q}, ’\overline{lx})$

,

satisfying

$\eta^{\prime/}(\overline{q},\overline{\tau x})-A_{\underline{)}}‘(t,,\overline{q})\eta’(\overline{q},\overline{\tau x})+[(A_{1}(t,,\overline{q})-A_{2}’(t,\overline{q}))+(N^{*}g_{y}(Ny(\overline{q},\overline{\uparrow x})N)^{*}]r)(\overline{q}, ’\overline{(x})$

$=C_{1}^{*}\Lambda_{M}(C’ 1^{(}’,/(\overline{q},\overline{\tau x})-z_{d})$

$\mathit{7}\mathit{1}(\overline{q}, ’\overline{l/})(T)=0,$

Since


$’/’(\overline{q},\overline{\tau x})(T)=0$

,

.

$C_{1}^{*}\Lambda_{M}(C_{1}y(\overline{q},\overline{\tau x})-z_{d})\in L^{\mathit{2}}‘(0, T;V_{2}^{*})$

and

$A_{2}’(t,,\overline{q})\in L^{\propto)}(0, T;\mathcal{L}(V_{2}, V_{2}^{*}))$

(4.5) is well-posed and permits a unique weak solution

the change of the time variable as

$farrow T-7$ .

$\eta(\overline{q},\overline{u})\in W(0, T)$

,

the equation

if we consider

Multiplying (4.5) by , which is the weak

solution to (4.2), integrating it by parts after integrating it on

$z$

$[0, T]$

, we obtain

$\int_{()}^{T}\langle\eta(\overline{q}, ’\overline{t/})(t), z^{\prime/}(t)+A_{\underline{)}}‘(t,\overline{q})z’(t)_{\mathrm{t}}[A_{1}(t,,\overline{q})+N^{*}g_{y}(Ny(\overline{q}, ’\overline{(x})(t))N]z(t,)\rangle_{VV}..dt$

(4.6)

$= \int_{()}^{T}\langle 77(\overline{q},\overline{?x})(f), -DA_{2}(t,\overline{q};q-\overline{q}).y’(\overline{q},\overline{u})(\dagger)-DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t)\rangle_{V^{*}.V}dt$

$+ \int_{(\}}^{T}\langle\eta(\overline{q}, ’\overline{lx})(t,), B(u-\overline{\tau x})(t)+f_{q}(t,,\overline{q};q-\overline{q})\rangle_{\mathfrak{l}\cdot,V}d\dagger\geq 0,$

$\forall(q, u)\in Q_{\tau}\cross[’$

.


From (4.3) and (4.4), we obtain the inequality $\int_{0}^{T}\langle\eta(\overline{q},\overline{\prime(x})(t), z^{\prime/}(t)+A_{\underline{)}}‘(t,\overline{q})z’(t)+[A_{1}(t,\overline{q})+N^{\mathrm{t}:}g_{y}(N.y(\overline{q}, ’\overline{lx})(t))N_{\mathrm{J}^{\mathrm{I}}}^{\urcorner}z(t)\rangle_{1/1/}.*.\cdot dt$

$+ \int_{()}^{T}\langle C_{1}y_{u}(\overline{q},\overline{u};q-\overline{q})(t,), C_{1}y(\overline{q},\overline{u})(t)-z_{d}\rangle d7$

$= \int_{()}^{T}\langle\eta(\overline{q},\overline{\tau x})(t), -DA_{2}(t_{J},\overline{q};q-\overline{q})y’(\overline{q},\overline{u})(t)-[)A_{1}(t,\overline{q};q-\overline{q})_{l/}’.(\overline{q}, ?\overline{x})(t)\rangle_{V^{*}.V}dt$

$+ \int_{0}^{T}\langle\eta(\overline{q},\overline{u})(t), B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{1V}.,dt$

$+ \int_{\mathrm{U}}^{T}\langle C_{1}y_{I4}(\overline{q},\overline{u};q-\overline{q})(t_{J}), C_{1}y(\overline{q},\overline{u})(t)-z_{d}\rangle_{V^{*}.V}dt\geq 0,$

[;.

$\forall(q, u)\in Q_{\tau}\cross$

Here we used the inequality (4.4). Summarizing these we have the following theorem.

Theorem 4.1. Assume $H(f)_{2},$

$H(g)_{1}$

$t,hafH(A),$

hold. Then the

$H(f),$

$H(B),$

$opt,imalco7\iota t,r‘$)

$H(N),$

$l(\overline{q},\overline{\uparrow x})\dot{u}$

$H(g),$

$H(A)_{1},$

charac $t,erized$ by

$H(A)_{2},$

staf,

$e$

a

$H(f)_{1}$

,

$7tdadjoi_{7}\iota t$

equations and inequality: $y”(\overline{q},\overline{u})+A_{2}(t,\overline{q})y’(\overline{q},\overline{u})+A_{1}(t,\overline{q})y(\overline{q},\overline{?x})+N^{*}g(N_{l/}’.(\overline{q},\overline{u}))=B\overline{?x}+f(t,\overline{q})$ $\{$

$y(\overline{q},\overline{u})(0)=y_{()}\in V,$ $y’(\overline{q},\overline{?x})(0)=y_{1}\in H$

$i\tau\iota,$

$(0, T)$

,

$\eta^{\prime/}(\overline{q},\overline{u})-A_{2}(t,\overline{q})\eta’(\overline{q},\overline{u})+[(A_{1}(t,\overline{q})-A_{2}’(t,\overline{q}))+(N^{*}g_{y}(N.y(\overline{q}, ’\overline{u})N)^{*}]\eta(\overline{q},\overline{u})$

$=C_{1}^{*}\Lambda_{M}(C_{1}y(\overline{q},\overline{u})-z_{d})$

$\{$

in

$(0, T)$

,

$\eta(T,\overline{q})=0,$ $\eta’(T,\overline{q})=0$

$\int_{\mathfrak{c})}^{T}\langle\eta(\overline{q},\overline{u})(t), B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{V^{*},V}dt$

$+ \int_{()}^{T}\langle C_{1}y_{u}(\overline{q},\overline{\tau x};u-\overline{?x})(t), C_{1}.y(\overline{q},\overline{u})(t)-z_{d}\rangle_{1^{r_{*}},V}/dt$ $\{$

$\geq\int_{(\}}^{T}\langle\eta(\overline{q},\overline{\prime u})(t), DA_{2}(\mathrm{r}_{J},\overline{q};q-\overline{q})y’(\overline{q}, \tau\overline{x})(t,)+DA_{1}(t,\overline{q};q-\overline{q})’.l/(\overline{q},\overline{7\mathrm{J}})(t)\rangle_{V^{*},V}dt$

,

.


$\square$

4.2. The case where

$C_{2}\in \mathcal{L}(H, M)$

In this case the cost functional is given by $J(q, u)= \frac{1}{2}||C_{2}y(q, u)(T)-z_{d}||_{M}^{2},$ $(q. n)\in Q_{\tau}\cross U$

and then the necessary condition (4.3) is equivalent to (4.7)

$(C_{2}^{*}\Lambda_{M}(C_{2}y(q, u)(T)-z_{d}),$ $z(T))_{H}$

$+(C_{2}^{*}\Lambda_{M}(C_{2}y(q, u)(T)-z_{d}),$

$y_{u}(\overline{q},\overline{u}:u-’\overline{(x})(T))_{H}\geq 0,$


.


Let us introduce an adjoint state

$\eta(\overline{q},\overline{u})\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\infty \mathrm{n}\mathrm{g}$

$\eta^{\prime/}(\overline{q},\overline{u})-A_{2}(t,\overline{q})\eta’(\overline{q},\overline{u})+\lfloor\lceil(A_{1}(t,\overline{q})-A_{2}’(t,\overline{q}))$

$+(N^{*}g_{y}(Ny(\overline{q},\overline{u})N)^{*}]\eta(\overline{q},\overline{\tau x})=0$

(4.8)

$\{$

$\prime\prime)(\overline{q},\overline{u})(T)=0$

,

$\eta’(\overline{q},\overline{u})(T)=-C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})$

.

It follows by the same reason as the case 4.1 that there is a unique weak solution $W(0, T)$

. because

$C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})\in H$

Theorem 4.2. We assume and

$H(f)_{2}$

adjoint,

hold.

$H(g)_{1}$

and

$equat,ions$

$t,hatH(A),$ $H(f),$

Then

$t,heopt,imal$

.

$H(B,),$ $H(N),$ $H(g),$

control


$H(A)_{1},$

$H(A)_{2},$

$H(f)_{1_{\rangle}}$

is characterized by state and

$inequalit,y$ :

$y”(\overline{q},\overline{u})+A_{2}(t,\overline{q})y’(\overline{q},\overline{u})+A_{1}(t,\overline{q})y(\overline{q},\overline{u})+N^{*}g(Ny(\overline{q},\overline{u}))=B\overline{u}+f\cdot(t, q)$ $\{$

$y(\overline{q},\overline{u})=y_{()}\in V,$

$\eta(\overline{q},\overline{u})\in$

$y’(\overline{q},\overline{u})=y_{1}\in H$

in

$(0, T)$

,

$\eta’’(\overline{q},\overline{u})-A_{2}(t,\overline{q})\eta’(\overline{q},\overline{u})+[(A_{1}(t,\overline{q})-A_{2}’(t,\overline{q}))+(N^{*}g_{y}(Ny(\overline{q},\overline{u})N)^{*}]\eta(\overline{q},\overline{u})(T)=0$

$\{$

$\eta(\overline{q},\overline{u})(T)=0$

,

,

$\eta’(\overline{q},\overline{\tau x})(T)=-C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{\prime lx})(T)-z_{d})$

$(C_{2}^{*}(C_{2}y( \overline{q},\overline{u})(T)-z_{d})_{H}+\int_{\{)}^{T}\langle B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q}), \eta(\overline{q},\overline{u})(t)\rangle_{V^{*},V}dt$

$\{$

$\geq\int_{()}^{T}\langle DA_{2}(t,\overline{q};q-\overline{q})y’(\overline{q},\overline{u})(t)+DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t), \eta(\overline{q},\overline{u})(t)\rangle_{V^{*},V}dt,$$\forall q\in Q_{\tau}$

.

PROOF. We prove the inequality condition of optimal control only. Multiplying (4.8) by , which is a weak solution to (4.2), integrating it by parts after integrating it on $z$

we obtain $\int_{()}^{T}\langle\eta(\overline{q},\overline{u})(t), z^{\prime/}(t)+A_{2}(t,\overline{u})z’(t)+[(A_{1}(t,\overline{q})+N^{*}g_{y}(Ny(\overline{q},\overline{u})(t)N]z(t)\rangle_{V^{*},V}dt$

$+(z(T), \eta’(\overline{q},\overline{\tau x})(T))_{H}$

$= \int_{()}^{T}\langle r/(\overline{q},\overline{?x})(t), -DA_{2}(t,\overline{q};q-\overline{q}).y’(\overline{q},\overline{u})(t)-DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t)\rangle_{VV}.,dt$

$+ \int_{()}^{T}\langle\eta(\overline{q},\overline{u})(t_{J}))B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{V^{*},V}dt$

$+(z(T), -C_{\underline{9}}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d}))_{H}=0$

.

$[0, t]$

.


Hence from (4.7) and (4.8) we conclude that $(z(T), C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d}))_{H}+(y_{u}(\overline{q},\overline{u};u-\overline{u})(T),$ $C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})_{H}$

$= \int_{0}^{T}\langle\eta(\overline{q},\overline{u})(t), -DA_{2}(t,\overline{q})q-\overline{q})y’(\overline{q},\overline{u})(t)-DA_{1}(t,\overline{q};q-\overline{q})y(\overline{q},\overline{u})(t)\rangle_{VV}.,dt$

$+ \int_{0}^{T}\langle\eta(\overline{q},\overline{u})(t), B(u-\overline{u})(t)+f_{q}(t,\overline{q};q-\overline{q})\rangle_{V^{*},V}dt$

$+(y_{\alpha}(\overline{q},\overline{u};u-\overline{u})(T),$

$C_{2}^{*}\Lambda_{M}(C_{2}y(\overline{q},\overline{u})(T)-z_{d})_{H}\geq 0,$

$(q, u)\in Q_{\tau}\cross U$

. $\square$

REFERENCES [1] N.U. Ahemd, Necessary conditions of optimality for a clas ‘9 of second order hyperbolic systems with spatially dependent controls in the coefficients, J. Optim. Theory and Appl., (1982), Vol. 38, pp.

423-446.

[2] N.U. Ahemd, Optimization and identification of systems govemed by $evol?4tion$ equabions on Banach space, Pitman research notes in Mathematics series, Longrrlan Scientific&Technical 184, 1988. [3] H.T. Banks, D.S. Gilliam and V.I. Shubov, Grobal solvability for damped abstmct nonlinear hyperbolic systems, Differential and integral equations, (1997), Vol. lt). No. 2, pp. 307-332. [4] H.T. Banks, K. Ito and Y. Wang, Well posedness for damped second order systems with unbounded input operators, Center for research in scientific computation, North Carolina state univ., 1993. , [5] H.T. Banks and K. Kunisch, Estimations Techniques for $d?str\cdot ibuted$ parameter systems. $\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{k}\mathrm{h}\ddot{a}\dagger \mathrm{l}\mathrm{s}\mathrm{e}\mathrm{r}$

1989.

and Tcchnol[6] R. Dautray and J.L. Lions, Mathemabicd Analysis and Numerical Methods for ogy, Springer-Verlag, 5, Evolution Problerns, 1992. [7] J.H. Ha and S.I. Nakagiri, Eristence and Regularity of solutions for sernil.inear second 07 der evolution equations, Funkcialaj Ekvacioj, (1998), Vol. 41, pp. 1-24. [8] J.H. Ha, Optimal control problems for hyperbolic distributecl parameter systems $e?thdamp?7\iota g$ terms. Doctoral Thesis, Kobe Univ., Japan, 1996. [9] J.L. Lions, Optimal control of systems governed by partial Differential $Equat_{?onS}$ , Springer-Verlag, New York, 1971. [10] O.A. Ladyzhenskaya, The boundary value problems of Mathematical physics, -Verlag, New York, 1984. . optirnality in on[11] A.Papageorigiou and N.S. Papageoriou, Neoessary and sufficient linear distributed parameter systems $w?th^{J}ua\dot{n}able$ initial stat , J. Math. Soc. Japan. (1990), Vol. 42, No. 3, pp. 387-396 [12] J.Y. Park, J.H. Ha and H.K. Han, Identification problem for damping pararnetars darnp $‘d$ second order systems. J. Korea Math. Soc., (1997), Vol. 34, No. 4, pp.1-14. [13] R. Temam, Infinite-Dimens?onal $Dynam?cal$ systems in Mechanic.9 and $phys?cs,$ -Verlag, New York, 1988. $\tau scien\mathrm{r}e$

$w\epsilon^{}ak$

$\mathrm{s}_{\mathrm{P}^{\mathrm{r}\mathrm{i}\mathrm{r}1}\mathrm{g}\mathrm{e}\mathrm{r}}$

$condit^{t}ionsf\dot{\mathrm{o}}r$

$\mathcal{T}1$

$‘$

$i_{7}\iota linea7^{\cdot}$

$\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{r}\iota \mathrm{g}\mathrm{e}\mathrm{r}$

DEPARTMENT

OF MATHEMATICS,

PIJSAN NATI

-mail address: [email protected]

$E$

$()\mathrm{N}\mathrm{A}\mathrm{L}\mathrm{I}\overline{\mathrm{I}}\mathrm{N}\mathrm{I}\mathrm{V}\mathrm{E}*\mathrm{I}\mathrm{T}\mathrm{Y}$

,

PUSAN

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